The riving knife on a table saw is a vital piece of safety equipment for the table saw. A riving knife is a flat piece of metal that is mounted just aft of the saw blade. When you push a board through the saw blade, the riving knife is designed to keep the two cut sections of the board from moving toward one another, thus pinching the saw blade and causing dangerous kickback. Even though a riving knife is effective in reducing the possibility of a kickback, it cannot prevent it entirely, which is why it is always advisable to follow the basic rules of use for a table saw: wear appropriate clothing, use properly approved safety glasses and hearing protection when necessary, keep your hands safely away from the saw blade, and stand in a manner so that, if the board does somehow kick back, you'll be less likely to sustain an injury.
On most table saws, the riving knife mounts to the saw's trunnion, which is the sturdy mechanism beneath the saw table that raises and lowers the saw blade. Typically, the riving knife has a release mechanism that can be accessed by removing the table insert from the top of the saw table. In many cases, this release mechanism allows the riving knife to be adjusted higher or lower in relation to the saw blade, and the knife typically is cut in a manner that there is less than 1/4 inch gap at any point between the cutting carbide teeth of the saw blade and the edge of the knife. This allows for maximum effectiveness when keeping the two halves of a freshly cut board separated as it is pushed through the blade.
When you watch many woodworking shows on TV, you'll often see that the riving knife, anti-kickback pawls and blade guards have been removed. While this is typically done to make viewing the steps involved easier for television viewing, it does a bit of a disservice to the novice woodworker in my mind, by telling woodworkers that it's OK to remove these safety devices.
At one time in my career, I felt that these safety devices were more of a hindrance than a help, but as time has gone on, not only has the technology improved, but I know that my attitude toward these safety devices has also softened to the point that I almost always use them when I can.
The key word in that last sentence is almost.
There are times when you simply must remove the riving knife, or when it is inappropriate to use one. A good example is when using a stacked dado blade to cut a dado or a rabbet. Because this type of blade does not create a through cut, a riving knife would serve no purpose in this instance. As a matter of fact, it would actually get in the way, blocking the completion of the cut. Additionally, since most table saws use a 10-inch diameter blade, but most stacked dado blades are 8-inches in diameter, there would be at least a one-inch gap between the edge of the stacked dado blade and the riving knife, which would essentially render the riving knife useless.
With all of that in mind, hopefully you can see the reasons for using a riving knife in almost every instance on your table saw. It is unobtrusive and certainly makes using your table saw safer.